Structure factor Debye term

In the procedure of X-ray refinement, the positions of the atoms and their fluctuations appear as parameters in the structure factor. These parameters are varied to match the experimentally determined strucmre factor. The term pertaining to the fluctuations is the Debye-Waller factor in which the atomic fluctuations are represented by the atomic distribution tensor ... 

Equation (24) shows that the response of the Fourier components of the density of a chain to a change of potential acting on that chain can be expressed in terms of the static structure factor SN(q) of the chain for gaussian chains this is simply the well-known Debye function [2]. 

This expression shows that the Go term acts as an overall Debye-Waller factor, as it would for any atom in the structure factor calculation, but that ow controls the overall behavior of this structure factor. If Gbar = 0, we reproduce the form of a truly crystalline CTR. We also recover the structure factor for an error function profile in the small-0 regime where this expression varies as l/iQcw, as long as 0ow 1. Therefore, this expression is capable of describing structures ranging from an error function profile associated with an disordered material to a highly layered fluid structure. 

The Rayleigh-Debye-Gans model of scattering predicts that for N particles in a scattering volume V, each particle will occupy a volume Vp and the total scattered intensity L from a beam of vertically polarized incident light is the summation of the individual scattering intensities, modified by an additional term that contains a structure factor S (rij) that accounts for the inter-particle interaction as shown in Eq. 2.35. 

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

Nj and Rj are the most important structural data that can be determined in an EXAFS analysis. Another parameter that characterizes the local structine aroimd the absorbing atom is the mean square displacement aj that siunmarizes the deviations of individual interatomic distances from the mean distance Rj of this neighboring shell. These deviations can be caused by vibrations or by structural disorder. The simple correction term exp [ 2k c ] is valid only in the case that the distribution of interatomic distances can be described by a Gaussian function, i.e., when a vibration or a pair distribution function is pmely harmonic. For the correct description of non-Gaussian pair distribution functions or of anhar-monic vibrations, different special models have been developed which lead to more complicated formulae [15-18]. This term, exp [-2k cj], is similar to the Debye-Waller factor correction used in X-ray diffraction however, the term as used here relates to deviations from a mean interatomic distance, whereas the Debye-Waller factor of X-ray diffraction describes deviations from a mean atomic position. 

In EXAFS experiments as well as in other EXAFS-like methods, variations in the sample temperature are well described by the Debye-Waller factor and lead to the exponential attenuation of line structure when the sample temperature increases. The temperature dependence of the SEFS spectrum is also described by the Debye-Waller factor— more exactly, by two Debye-Waller factors corresponding to the interference terms of the final and intermediate states [Eq. (38)). Since these interference terms are determined by different wave numbers, p and q, the change of the sample temperature results in a change of the relative intensity of the oscillating terms, which reveals itself in the unusual dependence behavior of SEFS. 

For example, Marshall and Slusher (1966) made a detailed evaluation of the solubility of ealeium sulphate in aqueous sodium chloride solution, and suggested that variations in the ion solubility product could be described, for ionic strengths up to around 2 M at temperatures from 0 to 100 °C, by adding another term in an extended Debye Hiickel expression. Above 2 M and below 25 °C, however, further correction factors had to be applied, the abnormal behaviour being attributed to an increase in the complexity of the structure of water under these circumstances. Enthalpies and entropies of solution and specific heat capacity were also reported as functions of ionic strength and temperature. 

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